Abstract

The method of matched asymptotic expansions is used to construct solutions for the planar steady flow of Oldroyd-B fluids around re-entrant corners of angles π/α (1/2≤α<1). Two types of similarity solutions are described for the core flow away from the walls. These correspond to the two main dominant balances of the constitutive equation, where the upper convected derivative of stress either dominates or is balanced by the upper convected derivative of the rate of strain. The former balance gives the incompressible Euler or inviscid flow equations and the latter balance the incompressible Navier–Stokes equations. The inviscid flow similarity solution for the core is that first derived by Hinch (Hinch 1993 J. Non-Newtonian Fluid Mech.50, 161–171) with a core stress singularity that depends upon the corner angle and radial distance as O(r−2(1−α)) and a velocity behaviour that vanishes as O(rα(3−α)−1). Extending the analysis of Renardy (Renardy 1995 J. Non-Newtonian Fluid Mech.58, 83–39), this outer solution is matched to viscometric wall behaviour for both upstream and downstream boundary layers. This structure is shown to hold for the majority of the retardation parameter range. In contrast, the similarity solution associated with the Navier–Stokes equations has a velocity behaviour O(rλ) where λ∈(0,1) satisfies a nonlinear eigenvalue problem, dependent upon the corner angle and an associated Reynolds number defined in terms of the ratio of the retardation and relaxation times. This similarity solution is shown to hold as an outer solution and is matched into stress boundary layers at the walls which recover viscometric behaviour. However, the matching is restricted to values of the retardation parameter close to the relaxation parameter. In this case the leading order core stress is Newtonian with behaviour O(r−(1−λ)).

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Footnotes

↵It is implicitly assumed that the limit ϵ→0+ is being considered with respect to which the asymptotic expansions are to be interpreted. Formal expansions in suitable gauges based upon the small parameter ϵ will be omitted when possible, in order to aid clarity and proliferation of notation.

↵This may be seen from equations (2.14)–(2.16) which have the exact solutionwhen λ2=λ1. This is the leading order Newtonian solution for the boundary layer, which with equation (2.13) shows that Ψ and are now inconsistent with the imposed matching conditions in equation (2.18). As such, the boundary layer similarity solution will fail to connect with the far-field behaviour as λ2 increases to λ1. Note also, that in the Newtonian case, necessarily T11 and T22 are the same order, the scales in equation (2.3) then implying that will be zero at both its leading and first-order terms (when considering a formal expansion in powers of δ/ϵ=ϵ1−α).

↵We note that this is not the most general solution to equation (1.6). We can generalize the solution associated with the upper convected stress derivative term in a manner similar to that described by Renardy (1997) for the UCM equations. However, the form presented here is anticipated to be the one of relevance at the re-entrant corner.

↵It is also worth remarking that the ‘density’ term (Re−G(ψ)) in equation (3.2) can, for n=1, retain the inertia terms. More generally, when G(ψ)=Re, the possibility of flows with zero or no pressure gradients arise for fluids with small retardation times. Such balances are of particular relevance to wedge flows of UCM fluids.

↵It may also be termed a ‘similarity parameter’ for the problem since it is only the combination c1λ1/λ2 that is important for classifying solutions.